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Formal Semantics in Modern Type Theories: Is It Model-Theoretic, Proof-Theoretic, or Both?

  • Zhaohui Luo
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8535)

Abstract

In this talk, we contend that, for NLs, the divide between model-theoretic semantics and proof-theoretic semantics has not been well-understood. In particular, the formal semantics based on modern type theories (MTTs) may be seen as both model-theoretic and proof-theoretic. To be more precise, it may be seen both ways in the sense that the NL semantics can first be represented in an MTT in a model-theoretic way and then the semantic representations can be understood inferentially in a proof-theoretic way. Considered in this way, MTTs arguably have unique advantages when employed for formal semantics.

Keywords

Type Theory Formal Semantic Proof Assistant Type Constructor Meaning Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Asher, N.: Lexical Meaning in Context: a Web of Words. Cambridge University Press (2012)Google Scholar
  2. 2.
    Asher, N., Luo, Z.: Formalisation of coercions in lexical semantics. Sinn und Bedeutung 17, Paris (2012)Google Scholar
  3. 3.
    Barwise, J., Perry, J.: Situations and Attitudes. MIT Press (1983)Google Scholar
  4. 4.
    Brandom, R.: Making It Explicit: Reasoning, Representing, and Discursive Commitment. Harvard University Press (1994)Google Scholar
  5. 5.
    Brandom, R.: Articulating Reasons: an Introduction to Inferentialism. Harvard University Press (2000)Google Scholar
  6. 6.
    Chatzikyriakidis, S.: Adverbs in a modern type theory. In: Asher, N., Soloviev, S. (eds.) LACL 2014. LNCS, vol. 8535, pp. 44–56. Springer, Heidelberg (2014)Google Scholar
  7. 7.
    Chatzikyriakidis, S., Luo, Z.: An account of natural language coordination in type theory with coercive subtyping. In: Duchier, D., Parmentier, Y. (eds.) CSLP 2012. LNCS, vol. 8114, pp. 31–51. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  8. 8.
    Chatzikyriakidis, S., Luo, Z.: Adjectives in a modern type-theoretical setting. In: Morrill, G., Nederhof, M.-J. (eds.) Formal Grammar 2012 and 2013. LNCS, vol. 8036, pp. 159–174. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  9. 9.
    Chatzikyriakidis, S., Luo, Z.: Natural language reasoning using proof-assistant technology: Rich typing and beyond. In: EACL Workshop on Type Theory and Natural Language Semantics, Goteborg (2014)Google Scholar
  10. 10.
    The Coq Development Team. The Coq Proof Assistant Reference Manual (Version 8.1), INRIA (2007)Google Scholar
  11. 11.
    Coquand, T., Huet, G.: The calculus of constructions. Information and Computation 76(2/3) (1988)Google Scholar
  12. 12.
    Dummett, M.: The Logical Basis of Metaphysics. Duckworth (1991)Google Scholar
  13. 13.
    Francez, N., Dyckhoff, R.: Proof-theoretic semantics for a natural language fragment. Linguistics and Philosophy 33(6) (2011)Google Scholar
  14. 14.
    Francez, N., Dyckhoff, R., Ben-Avi, G.: Proof-theoretic semantics for subsentential phrases. Studia Logica 94 (2010)Google Scholar
  15. 15.
    Gentzen, G.: Untersuchungen über das logische schliessen. Mathematische Zeitschrift 39 (1934)Google Scholar
  16. 16.
    Harper, R., Honsell, F., Plotkin, G.: A framework for defining logics. Journal of the Association for Computing Machinery 40(1), 143–184 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Kahle, R., Schroeder-Heister, P. (eds.): Proof-Theoretic Semantics. Special Issue of Synthese 148(3) (2006)Google Scholar
  18. 18.
    Luo, Z.: Computation and Reasoning: A Type Theory for Computer Science. Oxford University Press (1994)Google Scholar
  19. 19.
    Luo, Z.: Coercive subtyping. Journal of Logic and Computation 9(1), 105–130 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Luo, Z.: Manifest fields and module mechanisms in intensional type theory. In: Berardi, S., Damiani, F., de’Liguoro, U. (eds.) TYPES 2008. LNCS, vol. 5497, pp. 237–255. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  21. 21.
    Luo, Z.: Common nouns as types. In: Béchet, D., Dikovsky, A. (eds.) LACL 2012. LNCS, vol. 7351, pp. 173–185. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  22. 22.
    Luo, Z.: Formal semantics in modern type theories with coercive subtyping. Linguistics and Philosophy 35(6), 491–513 (2012)CrossRefGoogle Scholar
  23. 23.
    Luo, Z., Part, F.: Subtyping in type theory: Coercion contexts and local coercions (extended abstract). In: TYPES 2013, Toulouse (2013)Google Scholar
  24. 24.
    Luo, Z., Soloviev, S., Xue, T.: Coercive subtyping: theory and implementation. Information and Computation 223, 18–42 (2012)CrossRefMathSciNetGoogle Scholar
  25. 25.
    Martin-Löf, P.: Intuitionistic Type Theory. Bibliopolis (1984)Google Scholar
  26. 26.
    Martin-Löf, P.: On the meanings of the logical constants and the justifications of the logical laws. Nordic Journal of Philosophical Logic 1(1) (1996)Google Scholar
  27. 27.
    Montague, R.: Formal Philosophy. Yale University Press (1974); Collected papers Thomason, R (ed.)Google Scholar
  28. 28.
    Nordström, B., Petersson, K., Smith, J.: Programming in Martin-Löf’s Type Theory: An Introduction. Oxford University Press (1990)Google Scholar
  29. 29.
    Prawitz, D.: Towards a foundation of a general proof theory. In: Suppes, P., et al. (eds.) Logic, Methodology, and Phylosophy of Science IV (1973)Google Scholar
  30. 30.
    Prawitz, D.: On the idea of a general proof theory. Synthese 27 (1974)Google Scholar
  31. 31.
    Pustejovsky, J.: The Generative Lexicon. MIT Press (1995)Google Scholar
  32. 32.
    Ranta, A.: Type-Theoretical Grammar. Oxford University Press (1994)Google Scholar
  33. 33.
    Saeed, J.: Semantics. Wiley-Blackwell (1997)Google Scholar
  34. 34.
    Severi, P., Poll, E.: Pure type systems with definitions. In: Matiyasevich, Y.V., Nerode, A. (eds.) LFCS 1994. LNCS, vol. 813, pp. 316–328. Springer, Heidelberg (1994)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Zhaohui Luo
    • 1
  1. 1.Department of Computer ScienceRoyal Holloway, University of LondonUK

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